Actions of automorphism groups of free groups on homology spheres and acyclic manifolds

2008/03/13 by Bridson, Martin R., Vogtmann, Karen · 2 citations
#20F28 #20F6 #53C24 #57S25 #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.0803.2062

Abstract

For n at least 3, let SAut(F_n) denote the unique subgroup of index two in the automorphism group of a free group. The standard linear action of SL(n,Z) on Rⁿ induces non-trivial actions of SAut(F_n) on Rⁿ and on S^n-1. We prove that SAut(F_n) admits no non-trivial actions by homeomorphisms on acyclic manifolds or spheres of smaller dimension. Indeed, SAut(F_n) cannot act non-trivially on any generalized Z₂-homology sphere of dimension less than n-1, nor on any Z₂-acyclic Z₂-homology manifold of dimension less than n. It follows that SL(n,Z) cannot act non-trivially on such spaces either. When n is even, we obtain similar results with Z₃ coefficients.

Cited by

• Simple groups with strong fixed-point properties
• Helly-type theorems, CAT(0) spaces, and actions of automorphism groups of free groups

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